# Even numbers greater than 10 as sum of two specific odd numbers

It is well known fact that it is very hard to prove Goldbach's strong conjecture but perhaps some weaker variations can be proved(or disproved) ,so my question is: Is it true that every even number greater than 10 can be represented as the sum of an odd prime number and an odd semiprime?

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@yunone,Thanks... –  pedja Sep 13 '11 at 8:58
Have you seen Chen's theorem (en.wikipedia.org/wiki/Chen%27s_theorem) ? I don't know if that can be extended to just the prime + semiprime case, though –  yatima2975 Sep 13 '11 at 9:01
@yatima2975: That's probably as good an answer as this question will get here. (I was going to post it myself before I rechecked the comments.) –  anon Sep 13 '11 at 9:05
I wrote a small c++ program that verified the conjecture for all even numbers less than 600k and larger than 10. So perhaps it's true. –  JSchlather Sep 13 '11 at 9:12
math.utoledo.edu/~jevard/Page015.htm Has some references regarding improvements on Chen's theorem. –  JSchlather Sep 13 '11 at 11:57

## 1 Answer

Some counterexamples: 12, 14, 16, 30. My perl program can't find any more smaller than 100000.

EDIT: I didn't know that semiprimes are defined to include squares. When I comment out the line that filters them, nothing is output up to 100000. I'll leave this answer here as an example of wrongness.

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9+3=12, 9+5=14, 9+7=16, 25+5=30. Are you forgetting that $p^2$ is semiprime in your code? –  JSchlather Sep 13 '11 at 9:16
@Dan,12=3+9,9 is odd semiprime ;14=5+9 ;16=7+9; 30=5+25 ,25 is odd semiprime...so you are not right this time –  pedja Sep 13 '11 at 9:18